Oliver Fortmeier

A parallel strategy for a level set simulation of droplets moving in a liquid medium

The simulation of two-phase flow problems involving two time-dependent spatial regions with different physical properties is computationally hard. The numerical solution of such problems is complicated by the need to represent the movement of the interface. The level set approach is a front-capturing method representing the position of the interface implicitly by the root of a suitably defined function. We describe a parallel adaptive finite element simulation based on the level set approach. For freely sedimenting n-butanol droplets in water, we quantify the parallel performance on a Xeon-based cluster using up to 256 processes.

Parallel re-initialization of level set functions on distributed unstructured tetrahedral grids

Level set functions are employed to track interfaces in various application areas including simulation of two-phase flows and image segmentation. Often, a re-initializing algorithm is incorporated to transform a numerically instable level set function to a signed distance function. In this note, we present a parallel algorithm for re-initializing level set functions on unstructured, three-dimensional tetrahedral grids. The main idea behind this new domain decomposition approach is to combine a parallel brute-force re-initializing algorithm with an efficient way to compute distances between the interface and grid points. Time complexity and error analysis of the algorithm are investigated. Detailed numerical experiments demonstrate the accuracy and scalability on up to 128 processes.

Discrete and continuous adjoint approaches to estimate boundary heat fluxes in falling films

A wavy falling film simulation is considered in which a liquid travels along one side of a thin metal foil that is heated electrically from the opposite side. The direct problem consists of a three-dimensional heat conduction equation on a cuboid domain representing the foil with suitable initial and boundary conditions. The inverse problem consists of determining the heat flux on the film side of the foil from a given distribution of the temperature on the heating side. Two different adjoint approaches for the solution of this inverse problem are compared. In the continuous adjoint approach, the adjoint problem is analytically derived from the direct problem and then discretized. In the discrete adjoint approach, the direct problem is discretized, from which an adjoint code is generated by means of the reverse mode of automatic differentiation. Numerical experiments are reported, demonstrating the advantages and disadvantages of the two approaches.

Solving a parameter estimation problem in a three-dimensional conical tube on a parallel and distributed software infrastructure

Abstract A parameter estimation problem arising from a three-dimensional computational fluid dynamics problem is formulated. The goal is to estimate the inflow velocity of a fluid flowing through a conical measuring cell from actual nuclear magnetic resonance measurements of the velocity in a certain region of the flow field. The flow is described by the incompressible Navier–Stokes equations and numerically solved by the parallel adaptive finite element package DROPS. Three different optimization algorithms for the solution of the parameter estimation problem are compared from within the EFCOSS environment. In this distributed component-based software architecture, one can access various optimization algorithms without the need to manually adapt the interfaces between the parallel flow solver DROPS and the different serial optimization software packages. Numerical experiments on a cluster consisting of Xeon-based quad-core processors connected by an InfiniBand network are reported. The results demonstrate a successful estimation of the inflow velocity by different optimization algorithms and a reduction of the time to evaluate the objective of the parameter estimation by a factor of roughly 27 using 40 processes.

A Graph Model for Minimizing the Storage Overhead of Distributing Data for the Parallel Solution of Two-Phase Flows

We consider a finite element method for the parallel solution of two-phase flow problems using a level set approach. Here, two systems of equations result from the discretization of the governing partial differential equations. Rather than investigating the solution of these systems, we focus on finding a data distribution for their assembly. We formulate a new combinatorial problem that minimizes the overhead in storage requirement to represent the systems while, at the same time, balancing the computational effort to assemble these systems in parallel. We model this problem by introducing a weighted undirected graph. We then transform the problem to a (standard) graph partitioning problem in which a weighted sum of certain edges is minimized subject to balancing a weighted sum of all vertices. Numerical experiments are carried out illustrating the feasibility of the new approach for an application using up to 512 processes of a cluster of quad-core processors.

Hybrid Distributed-/Shared-Memory Parallelization For Re-initializing Level Set Functions

The ever-increasing power of high-performance computers and advances in numerical techniques make possible the realistic study of two-phase flow problems in three spatial dimensions. Unfortunately, today, there is often still a gap between the design of numerical algorithms and the characteristics of the hardware on which the algorithms are executed. For the solution of a particular sub problem of a two-phase flow problem, we develop a numerical algorithm that aims to match the architecture of a cluster of nodes with multi-core chips. The algorithm is concerned with the re-initialization of level set function used to keep track of the interface between two phases of a fluid. It consists of a hybrid MPI/OpenMP parallelization strategy, using a domain decomposition approach on the outermost level of parallelization. On the inner level, a parallel region handles an individual sub domain. So, a domain decomposition approach based on MPI is combined with an OpenMP approach leading to a hybrid distributed-/shared-memory parallelization. Numerical experiments show that using such a hybrid strategy scales better than a pure MPI parallelization on two different Xeon-based clusters of quad-core processors using up to 1024 cores.

A hybrid parallel algorithm for transforming finite element functions from adaptive to cartesian grids

High-performance computing is often used in various fields of computational science and engineering where the availability of a large memory is crucial to represent given problems. For problems arising from the discretization of partial differential equations, adaptivity is an alternative approach to reduce the storage requirements. We present an illustrating example demonstrating the need for an algorithm change when a serial hierarchical data structure is parallelized. More precisely, we investigate this issue taking as example an algorithm to transform finite element functions defined on an adaptive grid into functions on a Cartesian grid. By combining distributed- and shared-memory parallelization, the resulting hybrid parallel algorithm involves two levels of parallelism. The performance of the parallel algorithm is shown to be highly problem dependent. There are certain problem instances where the speedup is satisfactory while there is a dramatic performance reduction for other problem instances. The numerical experiments are given for transformations resulting from three-dimensional computational fluid dynamic problems. The problem instances are given by up to 750 000 tetrahedra representing about 3 millions of unknown variables. The experiments are carried out on a Xeon-based cluster using up to 64 processors.